# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Sequential closedness of Boolean algebras of projections in Banach spaces

### Tom 167 / 2005

Studia Mathematica 167 (2005), 45-62 MSC: Primary 46E27, 47B40; Secondary 46B26, 46B42. DOI: 10.4064/sm167-1-4

#### Streszczenie

Complete and $\sigma$-complete Boolean algebras of projections acting in a Banach space were introduced by W. Bade in the 1950's. A basic fact is that every complete Boolean algebra of projections is necessarily a closed set for the strong operator topology. Here we address the analogous question for $\sigma$-complete Boolean algebras: are they always a sequentially closed set for the strong operator topology? For the atomic case the answer is shown to be affirmative. For the general case, we develop criteria which characterize when a $\sigma$-complete Boolean algebra of projections is sequentially closed. These criteria are used to show that both possibilities occur: there exist examples which are sequentially closed and others which are not (even in Hilbert space).

#### Autorzy

• D. H. FremlinDepartment of Mathematics
University of Essex
Wivenhoe Park
Colchester CO4 3SQ, United Kingdom
e-mail
• B. de PagterDepartment of Applied Mathematical Analysis
Faculty EEMCS
Delft University of Technology
Mekelweg 4
2628CD Delft, The Netherlands
e-mail
• W. J. RickerMathematisch-Geographische Fakultät